Davies-trees in infinite combinatorics

This short note, prepared for the Logic Colloquium 2014, provides an introduction to Davies-trees and presents new applications in infinite combinatorics. In particular, we give new and simple proofs to the following theorems of P. Komj\'ath: every $n$-almost disjoint family of sets is essentially disjoint for any $n\in \mathbb N$; $\mathbb R^2$ is the union of $n+2$ clouds if the continuum is at most $\aleph_n$ for any $n\in \mathbb N$; every uncountably chromatic graph contains $n$-connected uncountably chromatic subgraphs for every $n\in \mathbb N$.Comment: 8 pages, prepared for the Logic Colloquium 201

Functions tiling with several lattices

We study the problem of finding a function $f$ with ``small support'' that simultaneously tiles with finitely many lattices $\Lambda_1, \ldots, \Lambda_N$ in $d$-dimensional Euclidean spaces. We prove several results, both upper bounds (constructions) and lower bounds on how large this support can and must be. We also study the problem in the setting of finite abelian groups, which turns out to be the most concrete setting. Several open questions are posed.Comment: 16 page

Domes over curves

A closed piecewise linear curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve $\gamma$ in $\mathbb{R}^3$, there is a dome over $\gamma$, i.e. whether $\gamma$ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when $\gamma$ is a quadrilateral, thus giving a negative solution to Kenyon's problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular $n$-gons.Comment: 16 figure

Kaleidoscopical configurations

Let G be a group and X be a G-space with the action G × X → X, (g, x) → gx. A subset A of X is called a kaleidoscopical configuration if there is a coloring χ : X → k (i.e. a mapping of X onto a cardinal k) such that the restriction χ|gA is a bijection for each g ∊ G. We survey some recent results on kaleidoscopical configurations in metric spaces considered as G-spaces with respect to the groups of its isometries and in groups considered as left regular G-spaces

Domes over Curves

A closed piecewise linear curve is called integral if it is comprised of unit intervals. Kenyon\u27s problem asks whether for every integral curve γ in ℝ3, there is a dome over γ, i.e. whether γ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when γ is a quadrilateral, thus giving a negative solution to Kenyon\u27s problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular n-gons